QM-BE methods the effective Hamiltonian

The particular aspect we shall treat here is the generalization of the basic theory of these methods to QM descriptions of the solute. To do that, one has to introduce the concept of the Effective Hamiltonian (EH) for the isolated system, the fundamental equation to solve is the standard Schrddinger equation 

In the presence of a solvent treated in the ASC-BE framework, an external perturbation has to be taken into account the reaction field due to the apparent charges placed on the cavity surface. Such perturbation can be represented in terms of an electrostatic potential 

Both the wave function and the eigenvalue E will be modified with respect to [8.119], due to the presence of the solvent perturbation. 

An important aspect, until now not introduced, is that the solvent apparent charges, and eonsequently the reaetion operator V, depend on the solute charge, i.e., in the present QM fi amework, on the wave function they eontribute to define. This mutual interactions between P and induees a eomplexity in the problem whieh can be solved through the stan- 

In the standard original model the perturbation is limited to the electrostatic effects (i.e., the electrostatic interaction between the apparent point charges and the solute charge distribution) however, extensions to include dispersion and repulsion effects have been formulated. In this more general context the operator can be thus partitioned in three terms (electrostatic, repulsive and dispersive), which all together contribute to modify the solute wave function. 

The 1/2 factor in front of accounts for the linear dependence of the operator on the solute charge (i.e., the quadratic dependence on ). In a more physical description, the same factor is introduced when one considers that half of the interaction energy has been spent in polarizing the solvent and it has not been included in G. 

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For QM-MM methods it is assumed that the effective Hamiltonian can be partitioned into quantum and classical components by writing [9]... 

In our discussion the usual Born-Oppenheimer (BO) approximation will be employed. This means that we assume a standard partition of the effective Hamiltonian into an electronic and a nuclear part, as well as the factorization of the solute wavefunction into an electronic and a nuclear component. As will be clear soon, the corresponding electronic problem is the main source of specificities of QM continuum models, due to the nonlinearity of the effective electronic Hamiltonian of the solute. The QM nuclear problem, whose solution gives information on solvent effects on the nuclear structure (geometry) and properties, has less specific aspects, with respect the case of the isolated molecules. In fact, once the proper potential energy surfaces are obtained from the solution of the electronic problem, such a problem can be solved using the standard methods and approximations (mechanical harmonicity, and anharmonicity of various order) used for isolated molecules. The QM nuclear problem is mainly connected with the vibrational properties of the nuclei and the corresponding spectroscopic observables, and it will be considered in more detail in the contributions in the book dedicated to the vibrational spectroscopies (IR/Raman). This contribution will be focused on the QM electronic problem. 

As said before, the nonlinear nature of the effective Hamiltonian implies that the Effective Schrodinger Equation (1.107) must be solved by an iterative process. The procedure, which represents the essence of any QM continuum solvation method, terminates when a convergence between the interaction reaction field of the solvent and the charge distribution of the solute is reached. 

In the frame of the hybrid methods it must be computed by a QM method. The Schrodinger equation with the effective Hamiltonian Hjf has multiple solutions, which describe excited states of the R-system provided the M-system is frozen in its ground state. Electronic energy of the system in the state expressed by the wave function eq. (1.231), has the form [29,30] ... 

Basic Theory of QM/MM Methods. - The underlying theory of the QM/MM approach has been covered by many different authors,97,98 so only a brief overview is given here. The energy of the whole system, E, can be written in terms of an effective Hamiltonian, HeS, and the electronic wavefunction of the QM atoms, ij/ ... 

The function Eq. (2) is very general and does not correspond to the assumed wave function of the hybrid QM/MM method. First of all the numbers of electrons in the subsystems must be fixed to apply computational schemes to separate parts on the legal ground. Second, we assume that the M-subsystem is treated with use of the MM, i.e, the PES of the M-subsystem is evaluated without explicit invocation of its wave function. The parameters of the M-subsystem must be transferable, Le, applicable to combination with any i -subsystem and even in the absence of it. For this purpose we should use the wave function of the ground state of the effective Hamiltonian for the M-subsystem since it is in a certain sense close the wave function calculated without any i -subsystem [73]. Thus the required wave function is represented by the antisymmetrized product of... 

The inter/intramolecular potentials that have been described may be viewed as classical in nature. An alternative is a hybrid quantum-mechanical/classical approach, in which the solute molecule is treated quantum-mechanically, but interactions involving the solvent are handled classically. Such methods are often labeled QM/MM, the MM reflecting the fact that classical force fields are utilized in molecular mechanics. An effective Hamiltonian Hefl is written for the entire solute/solvent system ... 

A different approach to mention here because it has some similarity to QM/MM is called RISM-SCF [5], It is based on a QM description of the solute, and makes use of some expressions of the integral equation of liquids (a physical approach that for reasons of space we cannot present here) to obtain in a simpler way the information encoded in the solvent distribution function used by MM and QM/MM methods. Both RISM-SCF and QM/MM use this information to define an effective Hamiltonian for the solute and both proceed step by step in improving the description of the solute electronic distribution and solvent distribution function, which in both methods are two coupled quantities. There is in this book a contribution by Sato dedicated to RISM-SCF to which the reader is referred. Sato also includes a mention of the 3D-RISM approach [6] which introduces important features in the physics of the model. In fact the simulation-based methods we have thus far mentioned use a spherically averaged radial distribution function, p(r) instead of a full position dependent function p(r) expression. For molecules of irregular shape and with groups of different polarity on the molecular periphery the examination of the averaged p(r) may lead to erroneous conclusions which have to be corrected in some way [7], The 3D version we have mentioned partly eliminates these artifacts. 

The details on the operators introduced in the two schemes will be given below, here we only want to add that the addition of Henv to the solute Hamiltonian automatically leads to a modification of the solute wavefunction which has now to be determined by solving the effective Eq. (1-1). This can be done using exactly the same methods used for isolated molecules here in particular we shall mainly focus on the standard self-consistent field (SCF) approach (either in its Hartree-Fock or DFT formulation). Due to the presence of Hem the modified SCF scheme is generally known as self-consistent reaction field (SCRF). Historically the term SCRF has been coined for the QM/continuum approach but here, due the parallelism between the two schemes which will be made clear in the following sections, it will be used indistinctly for both. 

As for the QM/MM description also for PCM, non-electrostatic (or van der Walls) terms can be added to the Vent operator in this case, besides the dispersion and repulsion terms, a new term has to be considered, namely the energy required to build a cavity of the proper shape and dimension in the continuum dielectric. This further continuum-specific term is generally indicated as cavitation. Generally all the non-electrostatic terms are expressed using empirical expressions and thus their effect is only on the energy and not on the solute wave function. As a matter of fact, dispersion and repulsion effects can be (and have been) described at a PCM-QM level and included in the solute-effective Hamiltonian 7/eff as two new operators modifying the SCRF scheme. Their definition can be found in Ref. [17] while a recent systematic comparison of these contributions determined either using the QM or the classical methods is reported in Ref. [18]... 

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